A: No - a 32w T8 lamp would not be compatible with the ballast in fixture that runs the 40w T12 lamp.
A: to check if the ballast is failing you can place known working bulbs into the fixture. If these bulbs do not operate correctly it is an indication that there is a problem with the ballast.
A: Yes - this will fit the same socket, but this is a 5000K daylight, where your existing lamp is a 4100K Cool white, so it will be a slightly different color of light. Check out www.homedepot.com/p/303811442 for the cool white
A: The size is the same, 48" X T-8. The color temperature maybe different. Not sure about that.
A: If you are putting known working bulbs into a fixture and they are not lighting up properly then the ballast in the fixture has likely failed and would need to be replaced.